Problem: Suppose we have a plane $P$ defined by the transformation $T$ for $0 < u < v$ and $4 < v < 10$. $T(u, v) = (2v, 5u - v, -4u + v)$ What is the surface area of $P$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $ \int_4^{10} \int_0^v \sqrt{119} \, du \, dv$ (Choice B) B $ \int_4^{10} \int_0^v 2\sqrt{37} \, du \, dv$ (Choice C) C $ \int_4^{10} \int_0^v 5\sqrt{39} \, du \, dv$ (Choice D) D $ \int_4^{10} \int_0^v \sqrt{165} \, du \, dv$
Solution: Assume we have a surface $S$ parameterized by a transformation $T$. If we want to find the surface integral over $S$ of a function $f$, we can use the formula below to convert it into a familiar double integral. $ \iint_S f(T(u, v)) | T_u \times T_v | \, du \, dv$ Finding surface area using a surface integral means using $f(x, y, z) = 1$. In effect, we are saying that we only care about the scaling factor caused by the area element. Therefore: $A = \int_4^{10} \int_0^v |T_u \times T_v| \, du \, dv$ Now we need to find the magnitude of the area element. $|T_u \times T_v| = \sqrt{165}$ [Calculation] In conclusion, the surface area of $P$ is: $ \int_4^{10} \int_0^v \sqrt{165} \, du \, dv$